Crossover between non-Markovian and Markovian dynamics induced by a hierarchical environment

Non-Markovian evolution of an open quantum system can be induced by the memory effects of a reservoir. Although a reservoir with stronger memory effects may seem like it should cause stronger non-Markovian effects on the system of interest, this seemingly intuitive thinking may not always be correct. We illustrate this by investigating a qubit (a two-level atom) that is coupled to a hierarchical environment, which contains a single-mode cavity and a reservoir consisting of an infinite number of modes. We show how the non-Markovian character of the system is influenced by the coupling strength between the qubit and cavity and the correlation time of the reservoir. In particular, we found a phenomenon whereby the qubit Markovian and non-Markovian transition exhibits a anomalous pattern in a parameter space depicted by the coupling strength and the correlation time of the reservoir.

Figure 1

Configuration of the system plus a hierarchical environment: The qubit of interest is coupled to a single-mode cavity while the cavity is coupled to a reservoir.

Figure 2

Change of the NM with respect to $\kappa$ for different $\lambda$. From top to bottom, $\lambda$ changes from $0.2\Gamma$ to $\infty$.

Figure 3

Change of the NM with respect to $\lambda$ for (a) the model where a qubit is coupled to a hierarchical environment consisting of one cavity and one reservoir with a memory time of $\tau ={\lambda }^{-1}$, as shown in Fig. 1, and (b) the model where a qubit is coupled to a reservoir with coupling strength $\kappa$ and memory time $\tau ={\lambda }^{-1}$ of the reservoir. It is worth mentioning that the dashed and dotted lines in (a) and all three lines in (b) decrease exactly to zero, though it is not shown in the figure.

Figure 4

The NM of the qubit for different $\kappa$ and $\lambda$. The non-Markovian regime is colored while the Markovian regime is white. The dashed black lines are the contour lines of the NM. The diamond blue line is the curve of the threshold ${\kappa }_T\left(\lambda \right)$.

Figure 5

Evolution of the trace distance $D\left(t\right)$ when $\kappa =0.3\Gamma$ and the pair of initial states is ${\rho }_1\left(0\right)=|+\rangle \langle +|$ and ${\rho }_2\left(0\right)=|-\rangle \langle -|$, where $|\pm \rangle =\frac{1}{\sqrt{2}}\left(\right|e\rangle \pm |g\rangle )$.